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SOME  INVARIANTS  AND  CO  VARIANTS 
OF  TERNARY  COLLINEATiONS 


HENRY  BAYARD  PHILLIPS 


A   DISSERTATION 

Submitted  to  the  Board  of  University  Studies  of  the  Johns  Hopkins 

University  in  Conformity  with  the  Requirements 

FOR  the  Degree  of  Doctor  of  Philosophy 


Press  of 

The  new  Era  Printing  Company 

Lancaster,  Pa. 

U)OT 


SOME  INVARIANTS  AND  COVARIANTS 
OF  TERNARY  COLLINEATIONS 


BY 

HENRY  BAYARD  PHILLIPS 


A  DISSERTATION 

Submitted  to  the  Board  of  University  Studies  of  the  Johns  Hopkins 

University  in  Conformity  with  the  Requirements 

FOR  the  Degree  of  Doctor  of  Philosophy 


Prem  of 

TNC  NtW  EDA  PRINTINQ  OOMPAHY 
UNOASTER,  Pa. 

1907 


SOME  INVARIANTS  AND  COYARIANTS  OF  TERNARY 
COLLINEATIONS. 

Introduction. 

1.  The  analytical  basis  of  the  present  paper  is  the  form  of  Grassmann's 
Liickenausdruck  which  Gibbs  called  a  dyadic.  This,  as  the  sequel  shows,  is 
merely  a  general  bilinear  function  from  which  the  variables  are  omitted.  It 
may  then  represent  a  collineation  or  correlation  and  may  be  manipulated  prac- 
tically like  the  ordinary  symbolical  bilinear  form. 

Starting  with  this  as  a  basis,  the  object  is  in  the  next  place  to  give  an  inter- 
pretation by  means  of  the  invariant  theory  of  various  double  products  suggested 
by  Gibbs  and  incidentally  to  obtain  some  of  the  properties  of  the  invariants 
and  covariants  involved.  The  field  of  operation  is  plane  projective  geometry 
and  the  products  are  formed  according  to  the  combinatory  multiplication  of 
Grassmann. 

Finally,  in  the  third  part,  there  is  considered  a  skew  symmetric  function 
of  any  number  of  collineations  which  is  called  an  alternant.  It  is  a  combinant, 
linear  in  the  coefficients  of  each  collineation,  and  presenting  in  some  ways  for 
functions  of  two  sets  of  variables  properties  analogous  to  those  of  the  expres- 
sions resulting  from  the  combinatory  multiplication  of  linear  manifolds. 

Part  I.     Notation. 
I.   The  open  product  or  dyadic. 

2.  In  a  space  of  two  dimensions  a  sum  of  mixed  products  of  similar  construc- 
tion, each  containing  a  single  factor  x,  may  be  written  in  the  form 

A^xB^  +  A^xB^  +  A^xB^, 

where  the  dot  is  used  to  show  that  the  order  of  multiplication  is  from  left  to 
right.  A^ ,  B^  and  x  are  geometric  quantities,  points  or  lines  of  the  plane,  and 
all  products  are  formed  according  to  the  combinatory  multiplication.  This 
may  be  considered  as  resulting  from  the  operation  of  x  on  the  expression 

A,{yB,^A,{).B,  +  A,{)-B,, 

the  operation  consisting  in  placing  the  variable  x  in  the  parentheses.  This  last 
expression  is  an  example  of  what  Grassman  called  an  open  product.* 

*"Au8dehnung8lehre"  (1878),  p.  265. 


165252 


4         SOME   INVARIANTS    AND    (X) VARIANTS    OF   TERNARY   COLLINEATIONS. 

Gibbs  wrote  the  open  product  in  the  form 

and  from  the  nature  of  its  construction  called  it  a  dyadic.*     The  variable  is 
supposed  to  operate  on  the  dyadic  from  the  outside  and  so  give  as  result 

xA^  •  B^  +  xA^  ■  B^  +  xA^  •  B^ 
or 

A^- B^x  +  A^-B^x  +  A^- B^x 

according  as  x  is  used  as  prefactor  or  postfactor. 

In  the  present  paper  the  notation  of  Gibbs  will  be  used  and  combinatory 
products  will  be  represented  either  by  placing  the  letters  in  parentheses  or  by 
placing  a  bar  over  them.  It  is  found  convenient  to  use  the  parentheses  when 
the  product  reduces  to  a  scalar,  or  number,  and  in  all  other  cases  to  use  the  bar. 
Unless  otherwise  expressly  stated  the  variable  will  enter  the  dyadic  as  post- 
factor,  i,  e.,  the  dyadic  will  operate  on  the  variable.  From  analogy  with  the 
ordinary  symbolism  for  a  row  product  we  shall  write 

AB  =  ^,^j  +  A^B^  +  ^3  J53. 

It  is  to  be  observed  that  A^  and  B^  in  this  expression  have  a  definite  size  or 
intensity.     If  they  are  only  projectively  given  the  dyadic  will  have  the  form 

AB  =  \A^B^  +  \A^B^  +  \A^B,, 

where  the  X's  are  numbers  determined  when  definite  intensities  are  given  to 
A^  and  B^. 

3.  As  an  operator  the  dyadic  gives  a  linear  transformation  of  quantities  con- 
tragredient  to  B^.     For,  x  being  such  a  quantity,  since  (B^x)  is  a  number, 

A(Bx)  =  X,{B,x)A,  +  \{B,x)A,  +  \{B,x)A, 

which  as  a  function  of  A^  is  a  simple  manifold  involving  x  linearly. 

There  are  two  cases  of  present  interest.  When  A^  and  B^  are  contragredient 
we  have  a  collineation  ;  when  cogredient,  a  correlation. 

A  dyadic  of  the  form 

aa  =  \a^a^  +  \a^a^  +  X3a3a3, 

where  the  a's  are  points  and  the  a's  lines,  represents  a  point  collineation. f     In 

*GlBB8'8  "Vector  Analysis"  (Wilson),  chap.  V. 
t  In  the  notation  of  Clebsch  this  is  of  course 

(af)(cur)  =  2A^(a<f)(aia;)=0, 

when  X  is  given  and  f  variable.     If  f  were  given  and  x  variable  the  dyadic  would  be  written  aa. 
The  dyadic  is  thus  regarded  not  as  giving  a  connex,  but  as  setting  up  a  definite  transformation. 


SOME   INVARIANTS    AND   COVARIANTS   OF   TERNARY   COLLINS ATIONS. 


particular  to  the  point  a^a^  corresponds  the  point 

To  the  vertices  of  the  triangle  a  then  correspond  the  points  a..  Since  the 
triangle  a  merely  presents  a  set  of  points  to  be  operated  upon  it  is  obvious 
that  this  may  be  chosen  at  random,  the  collineation  then  determining  a  as  its 
correspondent.  While  aa  as  an  operator  gives  the  collineation  of  points  it 
involves  internally  the  collineation  of  triads. 
Similarly  the  dyadic 

represents  a  correlation  in  which  the  lines  a.  correspond  to  the  points  of  the 
triangle  yS.     A  like  interpretation  may  be  given  for  the  dual  cases  aa  and  ab. 

II.   Tetrad  and  counter-tetrad. 

4.  We  have  seen  that  the  dyadic  in  trinomial  form  involves  the  correspond- 
ence of  triads.  Since,  however,  a  collineation  or  correlation  in  the  plane  is 
determined  by  four  pairs  of  corresponding  elements,  it  is  of  greater  interest  to 
have  the  dyadic  involve  a  correspondence  of  sets  of  four.  And  as  a  dyadic 
always  operates  on  a  contragredient  quantity  this  end  can  only  be  attained  by 
the  use  of  a  self  dual  scheme  of  four-point  and  four-line. 

With  a  4-point  we  associate  the  4-line  obtained  by  taking  the  polar  of  each 
point  with  respect  to  the  triangle  of  the  other  three.  It  is  well  known  then 
that  conversely  the  4-point  is  obtained  by  taking  the  polar  of  each  line  with 
respect  to  the  triangle  of  the  other  three.  These  mutually  related  systems  have 
been  called  tetrad  and  counter-tetrad.*^ 

Supposing  the  points  a.  to  satisfy  only  one  linear  relation  (which  is  the  only 
case  of  interest)  their  intensities  may  be  chosen  such  that 

(1)  a^  +  a2  +  a^+  a^  =  0. 

Operating  on  this  identity  with  the  products  a^a.  we  find  that  the  triple  products 
(a^a.aj^)  are  in  absolute  value  all  equal.     If  then  we  write 

we  obtain 

(2)  (a.a,.aj=±4  {i<j</c), 

where  the  sign  is  positive  or  negative  according  as  a.a.af^  is  complementary  to 
an  odd  or  an  even  term  in  the  sequence  a^a^a^a^. 

*  F.  MOELEY,  Trans.  Amer.  Math.  Society,  vol.  4,  p.  291. 


6         SOME    INVARIANTS   AND   COVARIANTS   OF   TERNARY   COLLINEATIONS. 

Making  use  of  these  formulas  the  counter-tetrad  a^  may  be  written  in  the 
canonical  form 


4a^  =  —  G^]  0^2  —  ^2^4  —  ^4^1> 


4a^  =      «!  Og  +  ajttg  +  03(1^. 
From  these  equations  by  addition  we  obtain 

(4)  aj  +  a2  +  a3+a^=0. 

Multiplying  the  equations  (3)  by  a.  and  making  use  of  (2)  it  is  easily  seen  that 

(5)  (a,a,)  =  3,  {a,a.)=-l  (i+j). 
We  have  here  sixteen  equations.     From  the  identity 

i  J 

it  is  seen  however  that  seven  of  these  equations  are  superfluous,  their  effect 
being  to  make  a^  and  a^  subject  to  conditions  (1)  and  (4).  When  one  tetrad  is 
given  there  are  then  nine  conditions  to  be  satisfied  by  the  other.  And  since  a 
tetrad  subject  to  the  conditions  (1)  or  (4)  in  addition  to  its  eight  geometrical 
constants  involves  an  undetermined  intensity  it  follows  that  there  is  a  single 
solution.  The  equations  (5)  may  therefore  be  taken  as  canonically  defining  a 
tetrad  and  counter-tetrad.  Their  symmetry  in  a^  and  a.  indicates  the  mutuality 
previously  mentioned. 

From  the  equations  (3)  by  direct  multiplication  we  obtain 

(6)  ^.=  a^-a^, 

where  i,jj  h,  /is  a  positive  permutation  of  the  numbers  ],  2,  3,  4.  Multiply- 
ing by  flj^  and  making  use  of  (5) 

(a,a.aj==±4  {i<3<k), 

the  sign  being  positive  or  negative  according  as  (^if^jOi,^  is  an  odd  or  even  minor 
of  a^a^a^a^.  From  the  symmetry  of  the  entire  system  in  a^  and  a.  we  may 
finally  write 

(8)  ^^.=  a,-a,, 

where  the  rule  of  subscripts  is  the  same  as  before. 

5.  The  application  of  the  preceding  to  the  study  of  dyadics  in  the  plane  is 
now  simple.     Consider  the  collineation 

(9)  E(«2«3«4)(^2/33/9J«,A. 


SOME   INVARIANTS  AND   COVARIANTS   OF  TERNARY  COLLINEATIONS.         7 

Taking    a^  and    /8.   subject    to    the   conditions    (2)   and   (7)    the    products 
(a^a.a^)(y8^yS./3j)  all  become  equal  and  (9)  becomes 

Operating  on  this  with  6^,  a  point  of  the  counter-tetrad  of  /8,  and  making  use 

of  (5)  we  get 

Sttj  —  ttj  —  ttg  —  a^  =  4a^ . 

The  points  h.  pass  by  (9)  into  the  points  a^ .     The  collineation  therefore  trans- 
forms the  associated  system  6,  /3  into  the  system  a,  a  and  so  the  dyadic  in  this 
form  involves  a  correspondence  of  tetrads. 
In  the  same  way  we  see  that 

(10)  Z(«2«3«J(^2^3^4)«l^I 

is  a  correlation  which  transforms  the  counter-tetrad  of  /S  into  a  and  so  carries 
the  system  b ,  yS  into  the  system  a,  a. 

Part  II.     Multiple  Products. 
I.  Multiple  products  are  complete  invariants. 

6.  With  two  dyadics  AA',  BB'  is  connected  a  form  ABA' B'  which  Gibbs 
called  the  double  product  of  the  two  dyadics.*  It  is  formed  by  multiplying 
the  dyadics  distributively,  each  pair  of  terms  combining  to  form  a  product  in 
which  the  prefactor  is  product  of  prefactors  and  postfactor  product  of  post- 
factors.  Gibbs  showed  that  this  double  multiplication  is  distributive  with 
respect  to  a  resolution  of  either  dyadic  or  is  invariantive  as  is  readily  seen  upon 
expansion. 

So  with  a  system  of  dyadics  are  a  series  of  multiple  products  given  by  the 
various  ways  in  which  prefactors  and  postfactors  may  be  independently  com- 
bined. From  their  construction  it  is  evident  that  such  forms  retain  their 
significance  when  the  prefactors  and  postfactors  are  transformed  separately  and 
therefore  belong  to  the  class  of  functions  that  Pasch  called  complete  invariants.f 
If  the  dyadics  appear  as  transformations  operating  on  the  elements  of  a  certain 
field,  since  a  transformation  of  postfactors  amounts  to  a  transformation  of  that 
field,  it  follows  that  the  geometric  interpretation  of  a  multiple  product  must 
involve  an  arbitrary  initial  field.  If,  for  example,  a  system  of  collineations  and 
correlations  in  the  plane  operate  upon  four  points,  the  multiple  products  will 
give  results  independent  of  the  initial  4-point,  i.  e.,  invariants  and  covariants 

*  Loo.  cit.,  p.  306. 

The  function  here  considered  is  a  double  product  only  in  the  sense  that  it  is  formed  by  a  cer- 
tain double  process.  It  is  neither  the  scalar  nor  the  vector  but  the  combinatorial  double  product. 
All  of  these  have  certain  properties  in  common  which  characterize  double  multiplication. 

t  "  Vollkommene  Invariante,"  Math.  Ann.,  Bd.  52,  p.  128. 


8        SOME   INVARIANTS   AND   COVARIANTS  OF  TERNARY  COLLINEATIONS. 

of  the  resulting  tetrads.     Illustrations  of  this  property  will  appear  in  the  dis- 
cussions that  follow. 

II.  Apolarity  of  coUineations. 

7.  Consider  two  contragredient  coUineations  aa  and  /86 ,  the  first  an  operator 
on  points,  the  second  an  operator  on  lines.  They  have  a  double  product  invar- 
iant (ay8)(a6).     When  this  vanishes  the  coUineations  will  be  called  apolar. 

To  see  the  meaning  of  this  write 

aa  =  \a,a^  +  \a^a2  +  \a^(^3, 

and  take  A6  =  Aa  as  reference  triangle.     That  is,  place 

(6,aJ=l,         (6,aJ  =  0. 
We  then  have 

(afi){ab)  =  \/^,(a,A)  +  \/*,(«2^2)  +  \f'si<^z^s)  -        . 

This  obviously  vanishes  when  for  each  value  of  the  subscript  a^  is  on  )8^ .  Two 
triangles  so  related  that  each  point  of  the  one  lies  on  the  corresponding  line  of 
the  other  will  be  called  incident.  Now  Aa  and  AyS  are  the  correspondents  of 
the  reference  triangle  with  respect  to  aa  and  yS6 .  Hence  apolarity  is  the  con- 
dition under  which  two  coUineations  can  send  a  triangle  into  a  pair  of  inci- 
dent triangles.* 

A  coUineation  apolar  to  the  identical  collineation  sends  certain  triangles  into 
incident  or  inscribed  triangles.     Such  a  collineation  has  been  called  normal. 

Write  the  given  collineation 

aa  =  XjOjaj  +  \a^a^  +  ^^s^s 

and  the  identical  collineation 

a^a^-\-a^a^+a^a^, 

where  Aa  =  Aa  is  reference  triangle.     The  apolarity  condition  is  then 

Hence  the  condition  for  normal  collineation  is  the  vanishing  of  what  Gibbs 
called  the  scalav^  of  the  dyadic. 

8.  Another  interpretation  for  apolarity  is  obtained  by  using  the  system  of 
counter-tetrads  explained  in  Art.  4.  The  invariant  (a/S)(a6)  gives  the  con- 
dition that  the  collineation 

a(a6)yS 

♦  M.  Pasoh,  Math.  Annalen,  Bd.  23,  p.  431. 
fLoo.  cii,  p.  275. 


SOME   INVARIANTS   AND   COVAEIANTS   OF   TERNARY   COLLINEATIONS.         9 

be  normal.  Hence  if  aa  and  ^b  operating  on  a  certain  tetrad  and  counter- 
tetrad  give  a  4-point  a^  and  a  4-line  yS^,  the  coUineation  which  transforms  6* 
(the  counter-tetrad  of  /3J  into  a.  will  be  normal.  According  to  (9)  the  coUinea- 
tion which  carries  6^  into  a^  is 

Therefore  the  condition  that  the  coUineations  aa  and  Bb  be  apolar  is 

(11)  Z(«2«3«4)(^2^3/34)(«lA)  =  0, 

where  a^  and  yS^  correspond  respectively  through  aa  and  y86  to  a  tetrad  of  points 
and  its  counter-tetrad  of  lines. 

It  is  to  be  observed  that  (11)  is  linear  in  each  of  the  quantities  a^  and  ^^. 
Hence  if  all  of  those  quantities  except  a  point  a^  are  given  it  will  lie  on  a  defi- 
nite line,  and  if  all  except  a  line  yS^  are  given  it  will  pass  through  a  definite 
point.  Therefore  a  4-point  and  4-line  subject  to  the  condition  (11)  determine 
a  tetrad  of  lines  through  a^  and  a  tetrad  of  points  on  /S^ ,  consisting,  in  fact,  of 
the  evectants  of  (11)  with  respect  to  a^  and  ^^. 

If  we  write  a^  and  /3,.  in  such  form  that  the  equations  (2)  and  (7)  hold,  the 
apolarity  condition  (11)  may  be  written 

Placing  {a^a^a^)  =  A ,  we  get  for  the  evectant  with  respect  to  a^ 


47l  ==  4^1  -  (a2^2)«3^4  +  {(^Z^i)^2^i  -  («4^4)«2«3- 

Placing  a^  as  counter-tetrad  of  a^  this  may  be  written  in  the  form 

47,  =  4  { /3,  -  i  i{aMa,  +  {a,^,)a^  +  {^M^z  +  («4^4)«4]  }  , 

as  is  readily  seen  upon  multiplying  the  right  hand  members  of  both  equations 
by  each  of  the  points  a^  and  using  the  identity 

(a^^J  +  {aM  +  (a3^3)  +  {aM  =  0, 

to  which  (11)  reduces.     Since  the  expression  in  brackets  is  symmetrical  with 
respect  to  the  numbers  1,  2,  3,  4  we  may  finally  write 

(12)  %  =  ^,-V 

where 

V  =  J  [(aiA)«l  +  («2^2)«2  +  («3/53)S  +  («4^4)«4]  ' 

From  (12)  and  (8)  we  have 
where  6  is  the  counter-tetrad  of  ^.     Hence  if  we  order  to  the  lines  7^  the 


10      SOME   INVARIANTS    AND   COVARIANTS   OF   TERNARY   COLLINEATIONS. 

points  6^  it  follows  that  each  pair  of  lines  intersect  on  the  join  of  the  remain- 
ing pair  of  points.     Such  a  4-point  and  4-line  may  be  called  chiastic.* 

From  the  symmetry  of  (11)  in  a^  and  y8^  we  are  now  able  to  write  the  evect- 
ant  with  respect  to  yS^.  in  the  form 

(13)  c.  =  a.-y 

where 

Therefore  the  four-point  c.  and  the  four-line  a.  are  chiastic. 

The  geometric  interpretation  of  apolarity  then  leads  to  the  following  state- 
ment of  a  theorem  of  Pasch  :  f 

Two  apolar  collineations  transform  any  four-point  and  associated  four-line 
into  a  four-point  a^  and  a  four-line  jS^  such  that  there  is  a  four-line  through  a»- 
chiastic  to  the  counter-tetrad  of  ^.  and  a  four-point  on  yS^  chiastic  to  the  counter- 
tetrad  of  a^ . 

From  (13)  it  is  observed  that  the  tetrads  c.  and  a.  are  perspective,  y  being 
the  center  of  perspective.  Since  counter-tetrads  are  chiastic  this  is  a  special 
case  of  a  theorem  of  Pasch  which  states  that  any  pair  of  four-points  chiastic 
to  the  same  four-line  are  perspective. 

9.  It  has  already  been  observed  that  the  apolarity  of  the  collineations  aa 
and  y86  amounts  to  the  vanishing  of  the  linear  invariant  (scalar)  of 

a{ab)^. 
For  brevity  we  shall  write 

s^  =  aa,  Sj  =  bfi, 

a^  =  aa,  o-^  =  /86. 

And  generally  we  shall  designate  a  collineation  by  s  and  its  inverse  by  a.  The 
collineation  written  above  is  then  Sj  Sj .  As  a  convenient  abbreviation  we  shall 
denote  the  linear  invariant  of  any  collineation  by  the  symbol  of  that  colline- 
ation placed  in  parentheses.  Thus  the  apolarity  of  aa  and  /86  is  given  sym- 
bolically by 

From  the  definition  it  follows  immediately  that 

(«i«2)  =  («2«i)  =  (o-iO-J  =  (o-2<r,). 

Similarly  we  shall  sometimes  find  it  convenient  to  denote  the  linear  invariant 

*Cf.  Sir  Robert  Ball,  "Theory  of  Screws,"  p.  306. 
t  Math.  Annalen,  Bd.  26,  p.  311. 

X  «i  and  8]  in  this  case  will  be  called  harmonic,  retaining  the  word  ai>olar  to  express  the  rela- 
tion of  «i  and  Of ,  or  «,  and  Oi . 


SOME   INVARIANTS   AND   COVAEIANTS    OF   TERNARY   COLLINEATIONS.      11 

of  the  product  of  any  number  of  collineations  Sj,  s,  •••,  s^  by  (s^s^-  •  -  s^). 
Writing  the  collineations  in  the  form  aa,  b^ ,cy,  etc.,  it  is  immediately  seen  that 

(14)  («1   ---VlS.)  =(«.«!   •••«r-l)- 

That  is,  the  linear  invariant  of  the  product  of  any  number  of  collineations  is 
not  affected  by  a  cyclic  permutation  of  those  collineations. 

10.  Suppose  we  have  three  collineations  each  of  which  transforms  a  3-line 
a.  into  a  3-line  /8^ .  If  a.  and  6^  are  the  points  of  the  triangles  a  and  fi  the 
collineations  may  be  written 

^2  =  /*1  A«X  +  f^2^2^2  +  /*3^3«8J 
0-3  =    I/,/3,a^  -I-    I'2^2«2  +    ^3^3  «3> 

where  \,  fi^,  and  y^are  all  different  from  zero.     Furthermore  let 

be  a  non-singular  coUineation  apolar  to  cr^,  a^,  and  0-3.     Taking  a  as  reference 
triangle  the  conditions  required  are 

PlM^^^i)  +  PM^2<^2)  +  Pz\i^^<^s)  =  0, 
PlM'l(fil^l)  +  P2P'2{^2^2)  +  PzP'A^Z^)  =  0, 
Px  "X  (  A^l)  +  P2  ^2(^2^2)  +  Pz  ""ii^A)  =  0. 

These  equations  can  only  coexist  when  either 


=  0 


or 

Pl(^lOl)  =  P2i^2'^2)  =  PlC/^sCs)  =  0. 

The  first  of  these  expresses  that  the  collineations  are  linearly  related  ;  the 
second  that  the  3-line  /3^  and  the  3-point  c^  are  incident.  Hence  if  a  non-singu- 
lar coUineation  is  apolar  to  three  linearly  independent  collineations  having  a 
common  triangle  pair,  it  transforms  the  first  of  those  triangles  into  one  incident 
to  the  second.  In  particular,  if  a.  and  ^.  are  identical,  their  triangle  is  trans- 
formed by  s  into  an  inscribed  triangle  and  is  consequently  a  Pasch  triangle* 
of  s. 

Suppose  a  second  set  of  collineations  T^ ,  T^  and  T^  which  transform  a  3-point 

*F.  MOELEY,  loc.  cit.,  p.  295. 


\ 

\ 

\ 

P'l 

P'2 

f^s 

^1 

^2 

^3 

12      SOME    INVARIANTS   AND   CO  VARIANTS   OP   TERNARY   COLLINEATIONS. 

d.  into  a  3-point  c. .     The  same  is  then  true  of  any  coUineation  of  the  net 

Suppose  one  of  these  coUineations  should  transform  a^  into  a  3-point  incident 
to  ^..     By  the  last  paragraph  the  conditions  required  are 

These  equations  may  be  satisfied  if 

{S,T,)     {S,T,)  (S,T,) 

(15)  {S,T,)     (S,T,)  {S,T,)  =0. 
(S,T,)     (S,T,)  {S,T,) 

From  the  symmetry  of  this  condition  in  S  and  T  we  conclude  that  if  there  is  a 
coUineation  which  transforms  d.  into  c.  and  a^  into  a  triad  incident  to  /8^.,  then 
there  is  a  coUineation  that  transforms  6,.  into  a^  and  c^  into  a  triad  incident  to  B^ . 
If  in  the  above  theorem  we  take  a^  equal  to  b^  and  c^  equal  to  d^  we  have  the 
theorem  of  Hun  that  the  relation  of  Pasch  triangle  and  fixed  triangle  in  a  nor- 
mal coUineation  is  mutual. 

III.   The  intermediate.* 

11.  Take  in  the  next  place  two  cogredient  point  coUineations  aa  and  6/3. 
They  have  a  double  product  aba^.-f  This  is  a  covariant  line  coUineation 
which  has  been  called  the  intermediate  of  aa  and  6/S . 

The  geometric  interpretation  is  easily  seen.  Let  r/  (Fig.  1)  be  the  correspon- 
dent of  any  line  f  two  of  whose  points  are  x  and  y  -  Then  by  a  well  known 
identity 

(16)  r}='^b{a^^)  =  ab{{ax)ifiy)-{ay){^x)}=¥^"-Wi' 

in  which  x'  and  x",  y'  and  y"  are  the  correspondents  of  a  and  y  with  respect  to 
aa  and  6y8.  Therefore  ?;  is  a  line  through  the  join  of  x'y"  and  x"y' j  or,  as  we 
may  say,  through  the  cross-join  of  the  correspondents  of  x  and  y.  Since  x  and 
y  are  any  points  on  ^,  ij  is  the  locus  of  cross-joins  of  pairs  of  points,  on  |.  This 
from  the  known  construction  of  a  polarity  amounts  to  saying  that  x'x"  envelopes 
a  conic  tangent  to  x'y'  and  x"y"  at  their  junction  with  17. 

*A.  B.  Coble,  Trans.  Amer.  Math.  Soc.,  vol.  4,  p.  70.  Prof.  Morley  has  used  the  word 
Clebschian  to  represent  a  form  of  this  kind  (IVans.,  vol.  4,  p.  471). 

t  In  the  symholic  notation  of  Clebsch  this  would  of  coarse  be  written 

(ate)(ay3f)=0, 
where  f  is  given  and  x  variable. 


SOME   INVARIANTS   AND   COVARIANTS   OF   TERNARY   COLLINEATIONS.      13 


In  case  of  three  points  jc ,  y ,  s  of  |  the  preceding  construction  involves  Pas- 
cal's theorem  for  the  hexagon  inscribed  in  a  two-line. 

In  case  the  two  coUineations  are  the  same,  rj  is  the  join  of  x  and  y'  and  the 
intermediate  reduces  to  the  reciprocal  or  line  form.     Hence  the  line  equation 


X 


V 


z 


Fig.  1. 

of  a  given  coUineation  is  gotten  by  taking  the  double  product  of  the  dyadic 
with  itself. 

12.  An  identity  for  which  we  shall  find  frequent  use  is  obtained  by  develop- 
ing (a6as)  (otySf )  according  to  the  ordinary  rule  for  multiplication  of  determi- 
nants.    We  thus  obtain 

{aa)     (a/3)     (a|) 

{ahx){a^^)=    (ba)      (b/S)     (6f) 

{xa)     (x^)     (jc|) 


14      SOME    INVARIANTS    AND   CO  VARIANTS   OF   TERNARY   COLLINEATIONS. 

=(ah^)(x^)+{a^){b^)(xa)  +  (a|)(6a)(x;S)  -  {aa){b^){xfi)  -  {b^){a^)(ax). 

Writing 

s^  =  aa,  §2  =  6)8 , 

and  using  the  notation  of  Art.  9  for  the  linear  invariants,  we  have 

(17)  «l«2=(«l«2)+<^l0"2  +  °"2°"l-(«l)0'2-(«2)<^l> 

where  the  bar  over  s^s^  is  used  to  signify  the  operation  of  forming  the  inter- 
mediate. 

13.  The  intermediate  belongs  to  a  type  of  correspondence  that  occurs  in  any 
number  of  dimensions.  And  though  we  are  at  present  concerned  with  the  col- 
lineation  in  the  plane  it  may  be  worth  the  trouble  to  indicate  the  general  theory. 

It  is  very  easy  to  extend  the  construction  already  given  for  the  plane  inter- 
mediate to  the  case  of  higher  dimensions.  Using  the  equations  (16)  we  have 
for  the  intermediate  ab  ayS  where  aa  and  bS  are  point  coUineations  in  space  that 
to  any  line  |  corresponds  a  linear  complex  with  reference  to  which  xy"  and  x"y' 
are  polar  lines,  x  and  x",  y  and  y",  being  correspondents  with  respect  to  aa.  and 
6y3  of  two  points  x  and  y  on^. 

In  the  same  way  for  the  intermediate  ahca^fy  we  have 

(18)  abG{a^y7r)  =  abc{(au){^yyz)  +  {ay){^yzx) -\-  (az){^y'xy)} 

where  x,  y,  z  are  three  points  of  the  plane  ir .  If  then  we  take  any  triangle  on 
the  plane  tt,  transform  its  points  by  the  collineation  aa,  transform  the  opposite 
lines  by  the  collineation  bc^y ,  and  join  the  corresponding  elements,  we  get  a 
set  of  three  planes  intersecting  on  the  correspondent  of  tt  with  respect  to  the 
collineation  abca^y.  If  bc^y  is  the  identical  line  collineation,  or  quadratic 
complex  to  which  every  line  belongs,  the  preceding  construction  is  readily 
translated  into  a  descriptive  property  of  two  complete  four-points  in  planes 
in  space. 

Proceeding  in  this  way  we  may  construct  intermediates  in  any  number  of 
dimensions.  Another  method  was  however  presented  by  Kraus  *  and  Muth.f 
Consider  in  the  first  place  the  plane  intermediate  written  in  the  form 

{abx){afi^)  =  0. 

This  expresses  the  apolarity  of  the  coUineations  axa^  and  bx^^.  For  on  form- 
ing the  double  product  we  get 

x{abx){afi^)^  =  0. 

♦"Dissertation"  (Giessen),  1886. 
■fMath.  AnncUen,  Bd.  33. 


SOME   INVARIANTS   AND   COVARIANTS   OP   TERNARY    COLLINEATIONS.      15 

Now  axa^  and  bx^^  may  be  considered  as  binary  projectivities  which  give  for 
points  on  |  lines  joining  x  to  the  correspondents  with  respect  to  aa  and  6y8. 
Then  since  two  binary  apolar  projectivities  give  rise  to  an  involution  it  follows 
that  if  we  transform  the  points  of  a  line  |  by  the  collineations  aa  and  6/3,  the 
correspondent  of  |  with  respect  to  a6a/3  is  the  locus  of  points  from  which  the 
transforms  appear  in  involution. 

Considering  x  and  |  as  lines  in  space  it  follows  from  the  argument  of  the 
last  paragraph  that  the  collineation  ab  a^  in  three  dimensions  gives  for  a  line  | 
the  complex  consisting  of  lines  which  joined  to  the  correspondents  of  points  on 
f  give  pairs  of  planes  belonging  to  an  involution. 

In  order  to  interpret  the  triple  intermediate  abooL^y  we  need  a  new  invariant. 
Three  plane  collineations  aa,  6/3,  and  cy  have  in  fact  a  triple  product  invariant 

{abc)(a^y). 

When  this  vanishes  the  collineations  have  been  called  harmonic*  Its  vanish- 
ing simply  expresses  that  the  intermediate  of  any  two  of  the  collineations  is 
apolar  to  the  third.  Since  we  are  able  to  construct  the  intermediate  and  to  inter- 
pret the  condition  of  apolarity  this  harmonic  relation  may  be  supposed  known. 
The  intermediate  of  three  collineations  aa,  6/3,  and  cy  of  space  may  be  writ- 
ten in  the  form 

(abcx)(  a^yir  )  =  0 . 

This  expresses  that  the  three  projectivities  ax  air,  bx^ir ,  and  cxyrr  are  har- 
monic.    For  on  forming  the  triple  product  we  have 

x(abcx)  {a^y7r)'7r  =  0. 

But  axaTT^  bx^ir ,  and  cxyjr  are  ternary  correspondences  which  give  for  points 
on  TT  lines  joining  x  to  their  correspondents  with  respect  to  aa,  b^,  and  07. 
Therefore  if  we  construct  with  respect  to  aa,  6/3,  and  ey  the  correspondents  of 
points  belonging  to  a  plane  tt,  the  correspondent  of  tt  with  respect  to  abcafiy 
is  the  locus  of  points  from  which  those  plane  systems  appear  harmonic. 

So  by  a  process  of  continuous  induction  we  may  build  up  intermediates  of 
any  degree  of  complexity.  An  intermediate  of  R  collineations  is  reducible  to 
a  harmonic  invariant  of  R  collineations  \n  R  —  1  dimensions.  This  may  be 
interpreted  as  the  apolarity  of  the  intermediate  oi  R  —1  collineations  and 
the  remaining  one.  Thus  given  the  knowledge  of  apolarity  the  intermediate 
of  jR  collineations  is  reducible  to  that  of  i2  —  1 . 

14.  A  collineation  is  singular  when  there  is  an  element  whose  correspondent 
is  indeterminate.  Thus  the  intermediate  aba^  in  the  plane  is  singular  when  a 
line  ^  can  be  found  such  that 

(19)  ^(a/8|)  =  0. 

*  J.  Keaus,  Math.  Annalen,  Bd.  29,  p.  234. 


16      SOME   INVAMANTS    AND    COVARIANTS   OF   TERNARY   CX>LLINEATIONS. 

From  the  construction  of  the  intermediate  it  is  evident  that  |  must  in  this  case 
pass  by  da  and  b^  into  the  same  line  77.  Now  (19)  is  the  condition  for  the 
correlation 

to  be  symmetrical,  i.  e.,  to  be  a  polarity.  But  a(a^^)b  sets  up  a  binary  corre- 
lation on  rj  consisting  of  pairs  of  points  given  by  aa  and  6/S  for  points  of  ^. 
Therefore  since  every  line  of  the  plane  cuts  | ,  it  follows  that  in  case  of  singular 
intermediate  every  line  of  the  plane  passes  by  aa  and  6/S  into  a  pair  of  lines 
apolar  to  a  definite  pair  of  points,  i.  e.,  the  double  points  of  the  binary  polarity 
on  the  correspondent  of  f , 

IV.  Apolarity  of  coUineation  and  correlation.* 

15.  A  coUineation  aa  and  a  contragredient  correlation  be  may  be  apolar,  i.  e., 
may  satisfy  the  condition  a6(ca)  =  0.  The  meaning  of  this  is  easily  seen. 
Write  them 

aa  =  \a^a^  +  X^a^a^  +  \«3«3> 

be  =  fi^b^G^  +  fM^b^c^  +  t^A^zy 

and  take  Ac  =  Aa  as  reference  triangle.     The  condition  of  apolarity  is  then 

ab{ca)=  \ti^aA  +  X^fi^a^  +  X^fM^aA  =  0. 

That  equation  expresses  that  the  triangles  a^  and  b^  are  perspective.  Therefore 
since  they  are  the  correspondents  through  aa  and  bo  of  the  reference  triangle, 
it  follows  that  a  coUineation  and  an  apolar  correlation  transform  any  triangle 
into  a  pair  of  perspective  triangles. 

Conversely  if  a^  and  b.  are  perspective  and  aa  is  given,  values  ^Ji.^  may  be 
found  such  that  (20)  holds.  Taking  those  values  as  the  coefficients  in  6c  we 
have  a  correlation  apolar  to  aa.  Therefore  if  a  coUineation  a/i  transform  the 
points  of  a  triangle  a  into  a  triad  perspective  to  b^,  there  is  a  correlation  apolar 
to  a/x  which  has  a^  and  b^  as  corresponding  pairs.  In  particular  a  coUineation 
and  correlation  are  apolar  if  they  transform  respectively  the  points  and  lines  of 
a  triangle  into  the  same  triad  of  points. 

The  condition  that  the  intermediate  ah  ayS  of  two  collineations  should  be  apo- 
lar to  a  correlation  cd  is 

{abc)a^d^  (abc)  {  a{^d)  -  fi{ad)  } 

=  bc{^d)aa^ac{ad)b^=0. 

Now bc{^d)  =  0,  and  ac ( ad)  =  0 ,  are  the  conditions  of  apolarity  of  6y9  and  aa 
with  cd.     Hence  if  two  collineations  are  apolar  to  a  correlation,  so  is  their  inter- 

*  F.  AscHiKBi  called  snoh  correspondences  harmonic.  Compare  his  article,  "  SuUe  omografle 
binarie  e  ternarie,"  Bend,  del  R.  Isiituto  Lonibardo,  (2 J  yoI.  24,  p.  289. 


SOME   INVARIANTS   AND   COVARIANTS   OF   TERNARY   COLLINEATIONS.      17 

mediate.  By  an  entirely  analogous  process  it  follows  that  if  two  correlations 
are  apolar  to  the  same  coUineation  their  intermediate  is  also. 

The  intermediate  of  a  coUineation  or  correlation  with  itself  is  the  reciprocal 
or  adjoined  form.  Hence  if  a  coUineation  and  correlation  are  apolar  the  same 
relation  subsists  when  either  or  both  are  replaced  by  their  adjoined  forms. 

If,  for  example,  two  coUineations  transform  the  points  of  a  triangle  a  into 
perspective  triads  we  have  seen  that  there  is  a  correlation  apolar  to  both  coUine- 
ations which  transforms  the  lines  a^  into  either  of  those  triads.  The  preceding 
paragraph  then  expresses  that,  if  two  coUineations  transform  a.  into  a  pair  of 
perspective  triangles,  the  intermediate  gives  a  triangle  perspective  to  both.* 

16.  A  correlation  apolar  to  the  identical  coUineation  transforms  any  triangle 
into  a  perspective  one  and  is  therefore  a  polarity.  The  condition  that  6c  be  a 
polarity  is  then 

(21)  6c=0. 

Suppose  a  coUineation  aa.  is  apolar  to  a  polarity  6c .     We  then  have 

a6(ac)  =  0, 

(22)  6^=0. 

Let  f  be  a  fixed  line  of  aa  given  by  the  root  \  of  the  characteristic  equation, 
i.  e.,  such  that 

(|a)a  =  X^ 

Multiplying  by  |  we  get  from  (22) 

(ac){(|6)a-(^a)6}  =  (ac)(^6)a  -  X(|c)6  =  0, 
(|c)6-(|6)c  =  0. 
Combining  these  equations  we  obtain 

(|c)(6a)a  =  X(|c)6. 

For  a  fixed  line  of  a  coUineation,  an  apolar  polarity  gives  a  fixed  'point  corre- 
sponding to  the  same  root  of  the  characteristic  equation.  If  the  characteristic 
equation  has  three  distinct  roots  the  fixed  triangle  is  then  self-conjugate  or 
polar  with  respect  to  any  apolar  polarity. 

From  (1 7)  we  have  for  the  adjoined  form  of  a  coUineation  s , 

(23)  ss  =  {is)  +  2a^  -  2{s)<7 , 

where  o-  is  the  inverse  of  s.     Since  a  polarity  is  symmetrical,  if  it  is  apolar  to 
5,  it  is  apolar  to  o-.     In  that  case  we  have  also  seen  that  it  is  apolar  to  »s  and 
*Cf.  MuTH,  Math  Annalen,  Bd.  40,  p.  98. 


18      SOME   INVARIANTS   AND   CO  VARIANTS   OF   TERNARY   COLLINEATIONS. 

identity.  Therefore  from  (23)  it  follows  that  if  a  polarity  is  apolar  to  s  it  is 
apolar  to  <7* .  All  collineations  that  are  co variants  of  a  are  however  expressible 
linearly  in  terms  of  o-",  o-  and  o^.*  Consequently,  a  'polarity  apolar  to  a  col- 
lineation  is  apolar  to  all  of  its  covariants. 

If  a  collineation  s  transforms  the  points  of  a  triangle  a  into  a  perspective 
triad,  there  is  a  polarity  which  transforms  the  lines  of  a  into  the  same  triad. 
That  polarity  is  obviously  apolar  to  s  and  to  all  of  its  covariants.  Further 
there  is  a  polarity  which  leaves  a  fixed  and  is  apolar  to  s.  Therefore  we  have 
Muth's  theorem  that  if  a  collineation  s  transforms  a  triangle  into  a  perspective 
one,  then  all  of  the  covariants  of  s  transform  it  into  triangles  perspective  to 
each  other  and  to  the  original  triangle.f 

17.  For  a  correlation  to  be  apolar  to  a  collineation  requires  the  identical 
vanishing  of  a  line  and  therefore  subjects  the  coefficients  of  either  to  three  linear 
conditions.  There  is  not  then  in  general  a  correlation  apolar  to  each  of  three 
given  collineations.  The  condition  for  such  is  the  vanishing  of  the  determinant 
of  the  nine  equations  expressing  the  three  apolarities.  This  invariant,  the 
explicit  form  of  which  does  not  concern  us,  has  been  called  A.J  It  is  a  com- 
binant  symmetrical  in  the  coefficients  of  the  three  collineations,  and  of  the  third 
degree  in  each.  We  will  now  consider  the  peculiarities  of  a  system  of  three 
collineations  for  which  A  vanishes. 

In  the  net 

(24)  S  =  Pl«l  +  P2«2+P3«5 

there  are  a  single  infinity  of  singular  collineations.  The  singular  points  lie  on 
a  cubic  that  we  may  call  C;  the  singular  lines  envelop  a  cubic  that  we  may 
call  r .  It  is  well  known  that  the  adjoined  form  of  a  singular  collineation  con- 
sists of  the  product  of  singular  line  and  singular  point.  And  we  have  seen 
that  a  correlation  apolar  tj  a  collineation  is  apolar  to  its  adjunct.  Therefore, 
if  a  correlation  is  apolar  to  the  collineations  s^ ,  s^  and  s^ ,  the  singular  lines  and 
points  of  (24)  are  correspondents  in  that  correlation  and  consequently  the  cubics 
r  and  C  are  reciprocal  through  it. 

With  a  point  of  C  is  associated  in  two  ways  a  line  of  F .  In  the  first  place 
the  point  a  appears  as  singular  point  in  a  definite  collineation  of  (24)  which  has 
a  singular  line  /3.  Secondly,  it  is  transformed  by  those  collineations  into  the 
points  of  a  definite  line  7 . 

To  say  that  a  correlation  is  apolar  to  each  of  three  collineations  amounts  to 
saying  that  those  collineations  operating  on  the  inverse  of  that  correlation  give 
polarities.  Such  a  transformation  of  the  collineations  does  not  afifect  the  cubic 
r  which  is  therefore  the  Cayleyan  of  the  three  polarities.     We  saw  above  how- 

*  Clebsch,  Vorlesungen  iiber  Geometric,  Bd.  1,  p.  991. 

fMuTH,  loo.  cit.,  p.  97, 

%  ROSANES,  Orelle,  Bd.  95,  p.  254. 


SOME   INVARIANTS   AND   CO  VARIANTS   OF   TERNARY   COLLINEATIONS.      19 

ever  that  the  line  yS  passes  by  the  inverse  of  the  apolar  correlation  into  the 
point  a .  Therefore  /S  passes  by  the  three  polarities  into  points  of  7  and  conse- 
quently j8  and  7  are  corresponding  lines  of  T . 

Conversely,  suppose  with  each  point  of  O  are  associated  a  pair  of  correspond- 
ing lines  of  V .  Two  coUineations^  of  (24)  whose  common  polar  triangles  are 
not  singular  may  be  written 

The  lines  yS^  are  singular  in  the  collineations  whose  singular  points  are  a^. 
The  points  ttj ,  a^,  a^  pass  by  the  net  of  collineations  respectively  into  points  of 


Fm.  2. 


20      SOME   INVARIANTS   AND   COVARIANTS   OF   TERNARY   CX)LLINEATIONS. 

7ij  72>  73  where  7.  is  incident  to  6..  By  supposition  /Sj7j,  P^'y^,  and  ^8373  are 
three  pairs  of  corresponding  lines  with  respect  to  the  curve.  If  we  should 
start  with  h^  we  could  construct  a  complete  four-point,  all  of  whose  lines  touch 
the  curve.  It  would  contain  those  three  pairs  of  lines  as  diagonal  pairs  and 
therefore  7i,  73,  73  pass  through  a  point  (Fig.  2).  The  triad  a^  then  passes  by 
two  of  the  coUineations  into  the  triad  h^  and  by  the  third  into  one  perspective 
to  6,..  According  to  Art.  15,  the  three  coUineations  therefore  have  a  common 
apolar  correlation. 

The  vanishing  of  the  invariant  A  is  the  necessary  and  sufficient  condition  that 
icith  any  point,  of  C  should  he  associated  (in  the  above  way)  a  pair  of  correspond- 
ing lines  of  V ,  and,  dually y  with  any  line  of  T  should  be  associated  a  pair  of 
corresponding  points  of  C. 

The  similarity  of  a  set  of  coUineations  for  which  A  vanishes  to  a  net  of 
polarities  is  noticeable.  It  is  due  to  the  fact  that  a  set  of  polarities  are  apolar 
to  the  identical  coUineation,  or,  what  amounts  to  the  same  thing,  that  a  net  of 
coUineations  with  vanishing  A  may  be  transformed  into  a  net  of  polarities  in 
such  a  way  as  to  have  either  the  initial  or  the  resultant  field  invariant. 

Part  III.     The  Alternant. 

I.  Introduction. 

16.  In  contrast  with  the  symmetrical  forms  just  considered  are  a  series  of 
combinants  that  we  will  call  alternants.  The  alternant  of  n  coUineations  s^  is 
defined  by  the  equation 


[«i  •••«„]  = 


where  the  determinant  is  supposed  to  be  developed  in  the  order  of  its  columns, 
i.  e.,  in  each  term  of  the  development  the  first  letter  is  taken  from  the  first  row, 
the  second  from  the  second  row,  etc.  This  determinant  is  readily  seen  to  follow 
the  ordinary  rules  so  far  as  its  rows  are  concerned.  If,  for  example,  a  linear 
relation  exists  between  the  coUineations  Sj,  •••,  s^,  the  alternant  is  zero. 
Using  the  ordinary  rule  of  signs,  the  determinant  may  be  developed  as  the  sum 
of  products  by  their  minors  of  determinants  of  rth  order  in  the  first  r  rows. 
The  alternant  cannot,  however,  in  general  be  developed  in  terms  of  its  columns. 
Obviously,  there  will  be  a  marked  difference  according  as  the  order  of  the 
alternant  is  odd  or  even.  If  the  order  n  is  even,  for  every  term  of  the  form 
s^-"  s.Sj^  will  be  a  term  —  «j«.  •  •  •  s^  where  the  intervening  letters  in  both  are 


h 

«2        • 

•        «» 

«1 

«2        • 

••    K 

«1 

«2        • 

•'    «„ 

SOME   INVAEIANT8   AND   CO  VARIANTS   OF   TERNAEY   COLLINEATIONS.      21 

the  same.     The  linear  invariant  is  therefore 

r{(Si---ssJ-(s4«i  •••«•)}. 

The  alternant  of  an  even  number  of  collineations  is  normal. 
Again  if  n  is  even  the  alternant  may  be  written  in  the  form 

(26)  T,{Ps,Q  -  Qs,P) 

where  P  and  Q  are  products  not  containing  s. .    This  is  harmonic  with  s.  *  since 

The  alternant  of  an  even  number  of  collineations  is  harmonic  with  each  of  them. 

Finally  from  (26)  placing  s^  to  be  identity  we  see  that  the  alternant  of  an  even 
number  of  collineations  vanishes  when  one  of  those  collineations  is  identity. 

In  case  of  an  odd  alternant,  since  the  members  of  a  cyclic  group  are  all  of 
the  same  sign,  we  have 

(27)  ([«i---«J)  =  '^(«x[«3---«„]). 

If  the  alternant  of  an  odd  number  of  collineations  is  normal,  each  collineaMon  is 
harmonic  vnth  the  alternant  of  the  remaining  n  —  1 . 
Write  the  alternant  in  the  form 

where  8^  is  the  minor  of  s,.  in  the  first  row  of  the  alternant.  If  n  is  odd  and 
one  of  the  collineations  «j  is  identity,  since  the  first  minors  are  even,  all  those 
containing  identity  vanish  and  the  alternant  takes  the  form 

[«1  •••««]    =«o(«2  •••««]    =    [«2---«n]- 

The  alternant  of  an  odd  number  of  collineations  containing  identity  is  equal  to  the 
alternant  of  the  remmning  n  —  1 . 

II.   The  binary  alternant. 

17.  The  alternant  [sjSg]  of  two  collineations  is  normal  and  harmonic  with 
each  of  the  collineations.  A  binary  normal  coUineation,  or  polarity,  is  apolar 
to  a  coUineation  s  only  when  the  fixed  points  of  s  form  a  pair  in  that  polarity. 
Therefore  [SjSg]  is  the  polarity  determined  by  the  common  harmonic  pair  of 
the  fixed  points  of  s^  and  s^  .f 

Consider  in  the  next  place  the  triple  alternant 


s. 


[«1«2«3] 


1  "2  "3 
1  ^2  ^3 
1        *2        ^3 


IJ^^ 


*  See  Art.  9,  footnote. 

t  Study,  "  Benare  Formen." 


22      SOME  INVARIANTS  AND  COVARIANTS   OF   TERNARY   COLLINEATIONS. 

It  is  normal  when  s^  is  harmonic  with  [SgSg]  .     Therefore  the  vanishing  of  the 
linear  invariant  of  [SiSj^s]  expresses  the  condition  that  the  fixed  points  of 
Sj,  Sj,  and  s^  should  lie  in  an  involution. 
The  condition  that  a  collineation 

(28)  ^  PiSl  +  p2^2  +  Psh 

be  harmonic  with  [SiSj^sl  ^^ 

0  =  Pl{(«l«2«3)-(«?«3«2)}    +P2{(«2«3«l)-(«2«1»3)}+P3{(«3»l«2)-(«3«2«1)}- 

Making  use  of  the  identity  «^  =  *s  —  A ,  this  becomes 

0  =   {  PlM  +  PiM  +  PsM}  {(«1«2«3)  -  («3«2«l)}  • 

The  alternant  is  therefore  harmonic  with  all  of  the  normal  coUineations  of  the 
pencil  pi8^  4-  ^2*2  +  Psh'  ^^  fixed  points  of  [s^s^s^']  give  the  coUineations  of 
(28)  whose  dyadics  are  squares. 

It  is  readily  seen  that  the  collineation 

s=  [s,s,s,]  - 'i  {{s^s.s^)  -  (s^s.s,)} 

is  harmonic  with  s^,s^,  and  s^.  When  s  is  identity  [s^Sg^s]  ^^  identity  and  con- 
versely. TTie  condition  that  s^,  s^,  and  s^  should  be  polarities  is  that  the  alter- 
nant l^s^s^s^'j  should  coincide  vnth  the  identical  collineation. 

When  [SjSjSj]  vanishes,  the  invariant  ( «j«2 S3 )  — (SjSgS^)  also  vanishes  and 
hence  s  is  zero.  The  three  coUineations  s^,  s^,  s^  have  no  definite  apolar  col- 
lineation and  therefore  satisfy  a  linear  identity.  The  vanishing  of  [SjSgSj]  is 
the  condition  for  the  three  coUineations  to  satisfy  a  linear  identity. 

From  any  four  linearly  independent  binary  coUineations  any  other  may  be 
linearly  derived.  In  particular  to  such  a  set  of  four  belongs  the  identical 
collineation.  Since  an  alternant  of  even  order  containing  identity  is  zero,  it 
follows  that  the  binary  alternant  of  fourth  order  vanishes  identically.  There- 
fore with  the  alternant  (s^s^^s^)  just  considered  the  discussion  in  the  binary 
domain  comes  to  an  end. 

III.    The  alternant  of  two  ternary  collineoiions. 
18.  We  shall  usually  write  the  coUineations  in  the  form 

Sj  =  aa , 

The  alternant  is  then 

[^1*2]  =  *i*2~*2*i  =  {ch)aP  —  {^a)ha. 

The  CO  variants  of  a  collineation  «  are  linear  functions  of  s^,  s,  and  «*,  where 
«j  is  the  identical  collineation.     Since  each  of  these  is  commutative  with  «  it 


SOME   INVARIANTS   AND   COVARIANTS   OF   TERNARY   COLLINEATIONS.       23 

follows  that  the  alternant  of  a  collineation  and  any  of  its  covariants  vanishes 
identically. 

The  invariant  relations  of  the  alternant  and  covariants  of  s^  and  s^  may  be 
summed  up  in  two  general  theorems. 

(i).   The  alternant  is  apolar  to  all  the  covariants  of  s^or  s^. 

For  let  cy  be  any  covariant  of  s^.     The  condition  of  apolarity  with  the 

alternant  is 

0  =  (a6)(a7)(/3c)  -  {Pa){hri){ac) 

=  6)S' [(07)00  —  (ac)7a]  , 

where  the  dot  is  used  to  represent  the  process  of  forming  the  double  product, 
or  bilinear  invariant.  Since  the  expression  in  brackets  is  the  alternant  of  aa. 
and  a  covariant  07,  the  function  vanishes  as  was  required. 

(ii).  Hie  alternant  is  apolar  to  the  intermediate  of  one  of  the  coUineations  and 
any  covariant  of  the  other. 

For  let  07  again  be  a  covariant  of  aa .     The  intermediate  of  this  with  b^  is 

60)87. 

The  condition  of  apolarity  with  the  alternant  is 

0  =  {ab'){abc){0'^y)  -  {^a){b'hG){a^y), 

where  b'^  is  a  new  symbol  for  6yS .  Interchanging  b^  and  6'y8'  and  adding, 
the  last  expression  becomes 


^  {{fi'/3y)(b'ba-  ca)  -  {b'bc){fi'^a  ya)}  =  l^jSb'b  ■  [yaca-ayac]  . 
The  expression  in  brackets  expands  into 

[aa)yG  —  (^ac)ya  —  (aa)yc  +  (^ay)ac  =  (ay)ac  —  [aG)ya, 
which  is  zero  since  it  is  the  alternant  of  aa  and  a  covariant. 

19.  Since  all  covariants  of  s  are  expressible  linearly  in  terms  of  5g,  s,s^,  the 
alternant  is  found  to  be  apolar  to  the  eight  coUineations 


(29) 


''OJ   <^1>    «■??   <^2>   O'L  «lSj>   «1«2^   «1«2> 


where,  as  in  Art.  9,  s^  s^  is  the  intermediate  of  s^  and  s^  and  o-  is  the  same  connex 
as  s  but  considered  reciprocally.  If  the  eight  coUineations  are  linearly  inde- 
pendent, the  eight  apolarity  conditions  are  sufficient  uniquely  to  determine  the 
alternant.  It  is  our  purpose  in  the  next  place  to  see  whether  or  when  such  is 
the  case. 

By  the  formula  (17)  we  have 

h4  =  {^l)  +  <^>l-^<^>l-(^l)<rl-(sl)<rl. 


24      SOME  INVARIANTS   AND   COVARIANTS  OF  TERNARY   COLLINEATIONS. 

Since  the  alternant  [s^Sj]  is  apolar  to  <r^,  tr^  and  al,  we  see  that  the  condition 
of  apolarity  of  [SiSj]  and  s^sl  is  the  vanishing  of  the  linear  invariant  of  the 
collineation 

Hence  by  direct  expansion  of  this  last  named  invariant  we  obtain 

,,^,         [«I«J  •«!«i  =  («1«2«1«2)  +  («1«2«D  -  («2«?«2)  -  («2«lS2«?) 

{6{J) 

=  («1«2«1«2)-(«2«1«2«?)> 

since  by  Art.  9  (SiSgSj)  and  {s^sls])  are  equal,  both  passing  by  a  cyclic  permu- 
tation into  (s'Sj). 
Again  we  have 

Since  [  s'  *i  ]  is  apolar  to  o-^ ,  c^  and  a^ ,  by  the  same  argument  as  before,  the 
apolarity  condition  of  [  Sj  «i  ]  and  si  s^  is  found  to  be  the  linear  invariant  of  the 
collineation 

{sls,-s,sl)(sls,  +  s,sl). 

Expanding  and  making  use  of  the  cyclic  permutation  we  then  obtain 

.„,,  [«2«l]  •  M2  =  («2«1«2)  +  («2«l«2«l)  -  («1«2«1«2)  "  i^A^l) 

[61) 

=  («l«2«f«2)-(«2«l«2«l)- 

In  like  manner,  making  use  of  the  identity 

M2  =  («1«2)  +  °"l<^2  +  <^2°"l  -(«l)°"2  -  («2)<^U 

we  obtain  the  apolarity  condition  of  [  s^  Sg  ]  and  s^  s.^  as  the  linear  invariant  of 

(«?«2-«2«l)(«l«2+«2«l) 

under  the  form 

(32)  [S^SI]   •M2=(«l«2«l«2)-(«2«l«2«?)- 

Since  [s^Sj]  is  a  normal  collineation  the  adjoined  form  is  given  by  (17)  as 


[«i«2]  [«i«2]  =  (  [«i«2]  [«i«2]  )  +  ^[o-xo-j'. 
The  discriminant  of  [SjSj]  is  then 


^l  =  K«l«2]   •  [«1«2]  [«1«2]   =  *[«1«2]   •   [<^l'^2]'  =  J(  [«1»2]') 
=  i  {(«1«2«1«2«1«2)  -  («l«2«l«2«l)  -  («1«2«N2)  "  ('2«f  «2«1»2) 

(33) 

-  (S2«,«2»i*j«i)  +   («2«i«2«l«2)  +  («2»l«2«l)  +  (  «I  «2  «1  «a«l )} 
=  («l«2«?«2)-(«2«l«2«f). 


SOME   INVAEIANT8   AND    CO  VARIANTS   OP   TERNARY    COLLINEATIONS.      25 

in  which  the  notation  A^^  is  used  to  show  that  it  is  the  discriminant  of  an  alter- 
nant involving  Sj  and  s^  each  to  the  first  degree. 

A  comparison  of  (30),  (31),  and  (32)  with  (33)  shows  that  the  various  invari- 
ants there  considered  are  equal  to  each  other  and  to  the  discriminant  of  [s^Sj]. 

Now  suppose  that  A^^  does  not  vanish  and  that  there  exists  a  linear  relation 
of  the  form 

(34)  Po^o  +  ^i<^i  +  P20"?  +  P3<^2  +  pA  +  P5M2  +  P6^2  +  P7M2  +  Ps^l  =  0- 

Since  according  to  (29),  [SjSg]  is  apolar  to  the  first  eight  collineations  in  that 
sequence  but  by  (30)  and  (33)  is  not  apolar  to  the  last,  it  follows  that  Pg  is  zero. 
Likewise,  operating  in  turn  with  [SjSg],  [«2*i]  and  [Si^z]?  ^®  ^^^  *^** 
Py  =  /Og  =  /jg  =  0 .     Hence  the  relation  (34)  must  be  of  the  form 

(35)  Po^o  ■^Pi'^i  +  P2<^1  +  PsS  +  Py2  =  0 
Replacing  a^  by  s^  and  forming  the  alternant  with  s,  we  obtain 

(36)  Pd^.s,]  +/>,[«,«!]  =0. 
Forming  the  bilinear  invariant  with  s[  $1 ,  this  gives 

Hence  p^=  0.  Likewise  on  operating  with  si  s^,  it  is  seen  that  p^  =  0.  Simi- 
larly p,  =  p^  =/>„=  0 . 

Hence  if  the  discriminant  Aj^  is  different  from  zero  no  linear  relation  of  the 
type  (34)  can  exist.  If  however  the  discriminant  is  zero,  since  the  nine  colline- 
ations are  apolar  to  [  «i  ^2  ]  >  ^^^J  must  satisfy  a  linear  relation.  Therefore,  the 
vanishing  of  the  discriminant  of  the  alternant  is  the  necessary  and  sufficient  con- 
dition for  the  existence  of  a  linear  relation  of  the  type  (34). 

Since  in  the  usual  case  the  discriminant  of  the  alternant  is  not  zero,  it  follows 
that  in  general  the  apolarity  conditions  of  (29)  are  independent  and  give  an  in- 
variant determination  of  the  alternant.  When  the  discriminant  is  zero,  all  the 
collineations  of  the  net 

(37)  ^[^1^2]    +\[«1«2]+\[*1«2]    +\[«^2] 

satisfy  those  conditions  and  the  determination  is  not  unique. 

20.  When  the  discriminant  vanishes  there  is  always  a  collineation  of  (37) 
that  vanishes  identically,  i.  e.,  the  four  alternants  satisfy  a  linear  relation.  For 
since  all  the  collineations  of  (37)  are  apolar  to  all  those  in  (34)  it  follows  that 
the  two  sets  must  contain  four  linear  relations.  If  one  of  these  belongs  to  (37) 
the  point  at  issue  is  settled.  If  not  there  must  be  four  equations  of  the  type 
(34).  Either  one  of  those  is  of  the  form  (35)  and  the  collineation  in  question 
is  (36),  or  it  is  possible  to  solve  for  one  of  the  intermediates  and  so  obtain 


26      SOME    INVARIANTS    AND    COVARIANTS   OF   TERNARY   COLLINEATIONS. 

an  equation 

Developing  s^s^  by  (17),  inverting,  and  forming  the  alternant  with  s^,  gives 

[5j«J    -(«l)[«l«2]   =\[«1«2]    +^4[Sl«2'] 

which  is  the  relation  desired. 

Suppose  conversely  that  the  four  alternants  satisfy  a  linear  relation 

In  this  equation  there  must  be  at  least  one  coefficient,  for  instance  X^ ,  that  is 
different  from  zero.  Operating  on  the  equation  with  s^sl  we  see  that  A^j  must 
then  vanish.  Therefore,  the  vanishing  of  the  discriminant  is  the  necessary  and 
sufficient  condition  for  a  linear  relation  between  the  four  alternants. 

21.  The  symmetry  resulting  when  Aj^  vanishes  suggests  that  it  is  an  invar- 
iant common  to  the  four  alternants.  In  order  to  prove  that  such  is  the  case, 
take  in  the  first  instance  the  alternant  [  «f  Sj  ] .  According  to  (33),  the  discrim- 
inant has  the  form 

^21  =  («lVt«2)--(«2«l«2«J)- 

From  the  characteristic  equation  for  s^  we  have 

4  =  \  +  \^i  +  \4' 

Substituting  this  value  for  s*  we  have 

(38)^      .  A,,=.\{(sls,s,sl)-{s,slsls,)]^-\A,,. 

Making  use  of 

«2  =  /*0  +  /*l«2  +  A*2«2 

and  following  out  the  same  argument  we  obtain  the  discriminants  of  [^i^j] 

and  [SjSj]  ^°  *^^  form 

(39)  |^.  =  -''.An. 

If  \j  is  zero  the  characteristic  equation  for  sj  is 

{sir  =  \  +  \sl. 

The  equation  is  quadratic  and  hence  the  collineation  is  a  perspectivity.  There- 
fore \  and  fij  are  respectively  the  invariants  whose  vanishing  expresses  that  sj 
and  si  are  perspectivities.  From  (38)  and  (39)  we  see  then  that  tlie  altemarU 
of  a  perspectivity  and  any  collineaiion  is  singular. 

If  Xj  and  ^l^  are  not  zero  Ajj  is  a  combinant  of  the  two  systems 
\Sq  4-  \j8j  +  X.2*i  ^^^  f^o^o  +  A''i^2  +  ^2*2-  -^^  ""^^^^  ^^®"  express  a  property  of 
the  fixed  triangles  of  those  systems.     What  that  property  is  we  shall  see  later. 


SOME   INVARIANTS   AND   COVARIANTS   OF   TERNARY   COLLINEATIONS.      27 

22.  The  alternant  [s^Sj]  ^^  ^  combinant  of  the  net 

(40)  \s^  +  fis^+vs^. 

In  forming  it  any  two  independent  collineations  may  then  be  chosen.  Let  s  be 
a  singular  collineation,  x  its  singular  point,  and  |  its  singular  line.  The  alter- 
nant may  be  written 

where  s^  is  some  other  collineation  of  the  net.  Since  sx  is  zero  the  correspondent 
of  X  with  respect  to  the  alternant  is 

Now  s  transforms  every  point  (and  in  particular  Sj  x)  into  a  point  on  f .  There- 
fore the  alternant  transforms  the  singular  point  of  any  collineation  of  (40)  into 
a  point  of  the  associated  singular  line.  For  varying  X,  the  singular  points  and 
lines  obtained  are  the  fixed  points  and  associated  fixed  lines  in  the  collineation 
/xSj  -|-  i/Sg.  Therefore  the  fixed  triangles  of  all  the  collineations  of  the  pencil 
fis^  +  vs^  are  JPasch  triangles  of  the  alternant. 

The  significance  of  the  apolarity  relations  satisfied  by  the  alternant  is  here 
suggested.  In  fact  two  collineations  (one  in  points,  the  other  in  lines)  that  send 
a  triangle  into  a  pair  of  incident  triangle  are  apolar.  Now  if  s  is  any  colline- 
ation in  (40)  the  triangle  that  s  leaves  fixed  is  sent  by  the  alternant  into  an 
inscribed  triangle.  Therefore  the  alternant  is  apolar  to  s  and  to  all  of  its 
covariants. 

Two  collineations  have  in  general  a  common  pair  of  polar  triangles.  In 
terms  of  these  they  may  be  written 

s^  =  /^i^iOCj  +  fi^\a^  +  fi^h^s- 
Their  adjoined  forms  and  intermediate  are  respectively 
«i«i  =  2  {  W^^a^  +  W^i^i  +  W^iCii  }  > 
«2«2  =  2  {  /^i^a/^a^a  +  ^2^3A«i  +  /*3/*i^2«2  }  f 

«1«2  =  {\f^2  +  \/*l)^3«3  +  (\/*3  +  \f^2)^l^l  +  {\f^l  +  \f^3)^2^2' 

Hence  s^^s^,  s^s^,  and  s^s^  have  a  common  pair  of  polar  triangles,  i.  e.,  the  com- 
mon pair  of  Sj  and  s^  considered  contragrediently.  Since  [s^Sg]  is  apolar  to 
each  of  those  collineations,  according  to  Art.  10,  it  transforms  the  triad  a^  into 
one  incident  to  /3^.  Taking  any  two  collineations  and  the  associated  interme- 
diate belonging  to  (29),  we  obtain  a  pair  of  polar  triangles  such  that  the  points 
of  the  first  pass  by  the  alternant  into  a  triad  incident  to  the  second.  These 
relations  are  therefore  the  geometric  equivalent  of  the  eight  apolarity  conditions. 


28      SOME   INVARIANTS   AND   COVAEIANTS    OP   TERNARY   <X)LLINEATIONS. 

23.  In  Arts.  19-21  the  entire  theory  seemed  to  hinge  on  the  vanishing  or 
non-vanishing  of  the  discriminant  of  the  alternant.  It  is  our  purpose  in  the 
next  place  to  consider  the  geometrical  interpretation  of  that  invariant. 

For  that  purpose  suppose  a  polarity  (?  to  be  apolar  to  s^  and  8^ .  Designating 
the  collineations  respectively  by  aa  and  bjS,  the  conditions  required  are 

(41)  ^(ac)  =  6^(^c)  =  0. 

There  are  six  equations  in  all,  three  of  them  linear  in  the  coefficients  of  aa, 
three  linear  in  the  coefficients  of  b^ ,  and  all  linear  in  the  coefficients  of  c^. 
Therefore  if  we  eliminate  the  coefficients  of  c^  from  these  equations  we  get  an 
invariant  of  the  third  degree  in  the  coefficients  of  aa  and  b^  whose  vanishing 
is  the  necessary  and  sufficient  condition  for  the  equations  (41). 

Now  we  have  seen  that  a  polarity  apolar  to  a  collineation  is  apolar  to  all  of 
its  covariants,  and  that  a  correlation  apolar  to  two  collineations  is  apolar  to 
their  intermediate.     Therefore  c^  is  apolar  to  the  following  nine  collineations, 

(42)  a^,  <7j,  0-2,  0-2,  0-2,  SjSj,  sfsj,  s^s],  s^sl. 

Since  the  nine  collineations  are  apolar  to  the  same  polarity  they  must  satisfy 
a  linear  relation,  in  fact,  must  satisfy  three  linear  relations.  Therefore  accord- 
ing to  Art.  19,  the  discriminant  of  the  alternant  is  zero.  But  the  discriminant 
is  of  the  third  degree  in  the  coefficients  of  aa  and  6/3.  Therefore,  the  vanish- 
ing of  the  discriminant  of  the  alternant  is  the  necessary  and  sufficient  conditions 
for  two  collineations  to  have  a  common  apolar  polarity. 

If  a  polarity  is  apolar  to  a  collineation  there  are  two  cases  to  be  considered 
according  as  the  polarity  is  singular  or  is  not. 

If  the  polarity  is  singular  it  either  consists  of  the  square  of  a  point  or  its 
adjoined  form  consists  of  the  square  of  a  line.  In  both  cases  if  the  polarity  is 
apolar  to  a  collineation  its  fixed  triangle  contains  the  double  element.  Hence 
if  a  singular  polarity  is  apolar  to  each  of  two  collineations  their  fixed  triangles 
have  an  element  in  common.  In  that  case  all  the  collineations  of  (42)  have  a 
fixed  point  or  line  in  common. 

In  general,  however,  if  a  polarity  is  apolar  to  a  collineation,  the  fixed  triangle 
(or,  a  fixed  triangle)  of  the  collineation  is  self-conjugate  with  respect  to  the 
polarity.  If  two  triangles  are  self-conjugate  with  respect  to  the  same  polarity 
they  lie  on  a  conic.  And,  conversely,  if  two  fixed  triangles  lie  on  a  conic,  they 
are  self-conjugate  with  respect  to  a  polarity,  which  is  consequntly  apolar  to  their 
associated  collineations.  Therefore  we  see  that  the  vanishing  of  the  discriminant 
of  the  alternant  is  the  necessary  and  sufficient  condition  for  any  two  collineations 
formed  linearly  from  those  of  (42)  to  have  fixed  triangles  lying  on  a  conic. 

24.  As  we  have  seen  the  conditions  for  a  polarity  c^  apolar  to  aa  and  6/8  are 

(ac)ac  =  (/Sc)6c  =  0. 


SOME   INVARIANTS    AND    COVARIANTS   OF   TERNARY    COLLINEATIONS.      29 

Multiplying  these  equations  by  6)8  and  aa  and  writing  in  the  variables  to  avoid 
confusion  we  get 

(ac){(/3c)(a|)-(^a)(c|)}(6^)^0, 

(/9c)  { («c)(6|)  -  («6)(c|) }  (ar;)  ^  0. 

Placing  I  and  77  equal  and  subtracting,  these  equations  give 

(43)  {(a6)(^c)(a|)-(^a)(ac)(6|)}(c|)  =  0. 

Comparing  this  with  the  general  form 

we  see  that  in  the  present  case  the  product  of  the  alternant  with  c^  is  a  correla- 
tion having  no  particular  coincidence  conic,  i.  e.,  a  nullsystem  or  line. 

Suppose  for  the  moment  we  designate  the  alternant  by  dh .     The  equation 

(43)  is  then 

(44)  (c^l)(Sc)(c|)  =  0, 

where  |  is  any  line  whatever.     Let  7;  be  a  fixed  line  of  the  alternant.     We 

then  have 

{d7))h  =  \r}, 

which  substituted  in  the  preceding  equation  gives 

(d7;)(8c)(c7/)  =  X(77c)2  =  0. 

Hence  the  fixed  lines  of  the  alternant  touch  the  conic  of  c^. 

If  again  we  follow  out  the  argument  of  this  article  with  7^,  the  adjoined 
form  of  c^,  we  obtain  as  correlative  to  (44) 

(45)  (7rf)(cca)(a;7)  =  0 

where  x  is  any  point  whatever.  Taking  cc  as  a  fixed  point  of  the  alternant,  it 
follows  as  before  that  the  fixed  lines  of  the  alternant  lie  on  7^. 

Now,  we  have  found  that  when  c^  is  apolar  to  s,  and  s^,  it  is  apolar  to  a  set 
of  nine  co variants.  Therefore,  all  the  alternants  that  can  he  formed  of  those 
Govariants  are  singular,  all  their  fixed  lines  are  tangent  to  c^  and  aU  their  fixed 
points  lie  on  7^ . 

It  was  shown  by  Study  that  the  fixed  points  of  the  alternant  of  two  binary 
coUineations  consist  of  the  common  harmonic  pair  of  the  fixed  points  of  those 
colli n cations.  The  fixed  triangles  of  two  ternary  coUineations  are  not  in  gen- 
eral polar  with  respect  to  the  same  conic.  If  such  however  is  the  case,  we 
have  just  seen  that  the  fixed  lines  of  the  alternant  are  tangent  to  and  the  fixed 
points  lie  on  that  conic. 


30      SOME   INVARIANTS    AND   <X>VARIANTS   OF   TERNARY    COLLINEATIONS. 


VITA. 

Henry  Bayard  Phillips  was  born  at  Yadkin  College,  N.  C,  September  27, 
1881.  He  received  his  preliminary  and  part  of  his  collegiate  education  at  that 
place.  In  1898  he  entered  Erskine  College,  Due  West,  S.  C,  and  graduated, 
B.  S.,  in  June,  1900.  After  teaching  for  one  year  at  Linwood,  N.  C,  he 
entered  the  Johns  Hopkins  University,  attending  courses  in  mathematics, 
physics,  and  philosophy.  There  he  was  successively  scholar  and  fellow  and 
graduated,  Ph.D.,  in  June,  1905. 


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